The mixed Hodge structure of an isolated critical point of a holomorphic function

Abstract

A mixed Hodge structure in a vector space is two filtrations of the space, satisfying the axioms indicated below. In the space of cohomologies, vanishing at the critical point of the holomorphic function, there is a natural mixed Hodge structure. The role of the above-mentioned filtrations is played by the weight and Hodge filtrations, introduced in Chapter 13. The weight filtration is constructed from the Jordan structure of the monodromy operator and reflects the behaviour of integrals over vanishing cycles under analytic continuation of the integrals round critical values of the parameter.

V.I. Arnold (deceased)